Weird little bathroom insulation questions
Hi, this is my first post, hopefully this makes some kind of sense.
I have a weird little house and an even weirder little bathroom. When entering the room the vanity is to the right on a wall that is only about 5'10". I am not sure why they built this part of the house with the roof coming so low down, but that's how tall the wall is. My girlfriend and I are only 5'6" and 5'4" so the low wall and mirror isn't really a problem. From the top of that wall the ceiling is vaulted for about 24" and them flattens out into a 2x4 framework so the ceiling is only around 6'4" and this goes across the room and into the shower. I don't know why the previous owner (who apparently fancied himself a real handyman) built the ceiling so low, I think it's to conceal the entire old fashioned vent fan above the ceiling. The ceiling framework at this low level does not contain any ceiling joists, it's just a weird framework made to support the sheetrock, so I can raise the ceiling at least to the point of whatever joists or collar ties are up there that I haven't discovered yet.
My plan is to vault the ceiling, but the rafters are not really deep enough to insulate it properly. My question is, would the following be a reasonable solution?
On the exterior walls (the short wall and the ceiling) I will use rigid foam insulation behind the sheetrock and foam batting in the rafters which will only be about 5 1/4". Then on the roof I will use another layer of rigid foam above the sheathing with 1x2 furring strips for vent space to a soffit vent and another layer of sheathing and then the final roofing with roof vents.
Whatever I do to this room I will also do to the other side of the roof so it is all insulated and vented the same way.
Hopefully this makes some kind of sense. I can't afford to knock it down and rebuild this part of the house which is what it really needs so hopefully this solution will be a sound way to build a bathroom. Or else it's back to the drawing board.